# On singular integrals of Calderón-type in $\mathbb R^n$, and BMO

### Dorina Mitrea

University of Missouri, Columbia, United States

## Abstract

We prove $L^p$ (and weighted $L^p$) bounds for singular integrals of the form

$\rm p.v. \int_{\mathbb R^n} E \lgroup \frac{A(x)–A(y)}{|x–y} \rgroup \frac{\Omega(x–y)}{|x–y|^n} f(y)dy,$

where $E(t) =$ cos $t$ if $\Omega$ is odd, and $E(t) =$ sin $t$ if $\Omega$ is even, and where $\bigtriangledown A \in$ BMO. Even in the case that $\Omega$ is smooth, the theory of singular integrals with "rough" kernels plays a key role in the proof. By standard techniques, the trigonometric function $E$ can then be replaced by a large class of smooth functions $F$. Some related operators are also considered. As a further application, we prove a compactness result for certain layer potentials.

## Cite this article

Dorina Mitrea, On singular integrals of Calderón-type in $\mathbb R^n$, and BMO. Rev. Mat. Iberoam. 10 (1994), no. 3, pp. 467–505

DOI 10.4171/RMI/159